Kinetic equation for a soliton gas - SOLITONS, COLLAPSES AND ...

N=3: explicit formulae (El, Kamchatnov, Pavlov & Zykov, 2008).The Riemann invariants r 1 , r 2 , r 3 are expressed in terms of the densitiesu 1 , u 2 , u 3 asr 1 =r 2 =r 3 =(ǫ 12 − ǫ 13 )(ǫ 12 ǫ 13 u 1 + ǫ 12 ǫ 23 u 2 + ǫ 13 ǫ 23 u 3 + ǫ 23 )[(ξ 3 − ξ 1 )ǫ 12 + (ξ 1 − ξ 2 )ǫ 13 ]u 1 − (ξ 2 − ξ 3 )(ǫ 12 u 2 + ǫ 13 u 3 + 1) ,(ǫ 23 − ǫ 12 )(ǫ 12 ǫ 13 u 1 + ǫ 12 ǫ 23 u 2 + ǫ 13 ǫ 23 u 3 + ǫ 13 )[(ξ 1 − ξ 2 )ǫ 23 + (ξ 2 − ξ 3 )ǫ 12 ]u 2 − (ξ 3 − ξ 1 )(ǫ 12 u 1 + ǫ 23 u 3 + 1) ,(ǫ 13 − ǫ 23 )(ǫ 12 ǫ 13 u 1 + ǫ 12 ǫ 23 u 2 + ǫ 13 ǫ 23 u 3 + ǫ 12 )[(ξ 2 − ξ 3 )ǫ 13 + (ξ 3 − ξ 1 )ǫ 23 ]u 3 − (ξ 1 − ξ 2 )(ǫ 13 u 1 + ǫ 23 u 2 + 1) .Gennady ElKinetic equation for a soliton gas